LMFDB - Embedded newform 4002.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -0.449490 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -0.449490 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -4.89898 q^{11} +1.00000 q^{12} +2.89898 q^{13} -0.449490 q^{14} +4.00000 q^{15} +1.00000 q^{16} +4.89898 q^{17} +1.00000 q^{18} +0.449490 q^{19} +4.00000 q^{20} -0.449490 q^{21} -4.89898 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +2.89898 q^{26} +1.00000 q^{27} -0.449490 q^{28} +1.00000 q^{29} +4.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} -4.89898 q^{33} +4.89898 q^{34} -1.79796 q^{35} +1.00000 q^{36} -2.00000 q^{37} +0.449490 q^{38} +2.89898 q^{39} +4.00000 q^{40} +0.898979 q^{41} -0.449490 q^{42} +7.55051 q^{43} -4.89898 q^{44} +4.00000 q^{45} -1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -6.79796 q^{49} +11.0000 q^{50} +4.89898 q^{51} +2.89898 q^{52} +0.898979 q^{53} +1.00000 q^{54} -19.5959 q^{55} -0.449490 q^{56} +0.449490 q^{57} +1.00000 q^{58} +4.89898 q^{59} +4.00000 q^{60} -6.89898 q^{61} -6.00000 q^{62} -0.449490 q^{63} +1.00000 q^{64} +11.5959 q^{65} -4.89898 q^{66} +2.00000 q^{67} +4.89898 q^{68} -1.00000 q^{69} -1.79796 q^{70} -1.10102 q^{71} +1.00000 q^{72} +2.89898 q^{73} -2.00000 q^{74} +11.0000 q^{75} +0.449490 q^{76} +2.20204 q^{77} +2.89898 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +0.898979 q^{82} -6.44949 q^{83} -0.449490 q^{84} +19.5959 q^{85} +7.55051 q^{86} +1.00000 q^{87} -4.89898 q^{88} +3.10102 q^{89} +4.00000 q^{90} -1.30306 q^{91} -1.00000 q^{92} -6.00000 q^{93} +4.00000 q^{94} +1.79796 q^{95} +1.00000 q^{96} +15.1464 q^{97} -6.79796 q^{98} -4.89898 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 8 q^{5} + 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 8 * q^5 + 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 8 q^{5} + 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9} + 8 q^{10} + 2 q^{12} - 4 q^{13} + 4 q^{14} + 8 q^{15} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 8 q^{20} + 4 q^{21} - 2 q^{23} + 2 q^{24} + 22 q^{25} - 4 q^{26} + 2 q^{27} + 4 q^{28} + 2 q^{29} + 8 q^{30} - 12 q^{31} + 2 q^{32} + 16 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} + 8 q^{40} - 8 q^{41} + 4 q^{42} + 20 q^{43} + 8 q^{45} - 2 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{49} + 22 q^{50} - 4 q^{52} - 8 q^{53} + 2 q^{54} + 4 q^{56} - 4 q^{57} + 2 q^{58} + 8 q^{60} - 4 q^{61} - 12 q^{62} + 4 q^{63} + 2 q^{64} - 16 q^{65} + 4 q^{67} - 2 q^{69} + 16 q^{70} - 12 q^{71} + 2 q^{72} - 4 q^{73} - 4 q^{74} + 22 q^{75} - 4 q^{76} + 24 q^{77} - 4 q^{78} + 20 q^{79} + 8 q^{80} + 2 q^{81} - 8 q^{82} - 8 q^{83} + 4 q^{84} + 20 q^{86} + 2 q^{87} + 16 q^{89} + 8 q^{90} - 32 q^{91} - 2 q^{92} - 12 q^{93} + 8 q^{94} - 16 q^{95} + 2 q^{96} - 4 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 8 * q^5 + 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9 + 8 * q^10 + 2 * q^12 - 4 * q^13 + 4 * q^14 + 8 * q^15 + 2 * q^16 + 2 * q^18 - 4 * q^19 + 8 * q^20 + 4 * q^21 - 2 * q^23 + 2 * q^24 + 22 * q^25 - 4 * q^26 + 2 * q^27 + 4 * q^28 + 2 * q^29 + 8 * q^30 - 12 * q^31 + 2 * q^32 + 16 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 + 8 * q^40 - 8 * q^41 + 4 * q^42 + 20 * q^43 + 8 * q^45 - 2 * q^46 + 8 * q^47 + 2 * q^48 + 6 * q^49 + 22 * q^50 - 4 * q^52 - 8 * q^53 + 2 * q^54 + 4 * q^56 - 4 * q^57 + 2 * q^58 + 8 * q^60 - 4 * q^61 - 12 * q^62 + 4 * q^63 + 2 * q^64 - 16 * q^65 + 4 * q^67 - 2 * q^69 + 16 * q^70 - 12 * q^71 + 2 * q^72 - 4 * q^73 - 4 * q^74 + 22 * q^75 - 4 * q^76 + 24 * q^77 - 4 * q^78 + 20 * q^79 + 8 * q^80 + 2 * q^81 - 8 * q^82 - 8 * q^83 + 4 * q^84 + 20 * q^86 + 2 * q^87 + 16 * q^89 + 8 * q^90 - 32 * q^91 - 2 * q^92 - 12 * q^93 + 8 * q^94 - 16 * q^95 + 2 * q^96 - 4 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	1.00000	0.408248
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	1.00000	0.288675
$13$	2.89898	0.804032	0.402016	−	0.915633i	$-0.368310\pi$
$13$	2.89898	0.804032	0.402016	+	0.915633i	$0.368310\pi$
$14$	−0.449490	−0.120131
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	1.00000	0.235702
$19$	0.449490	0.103120	0.0515600	−	0.998670i	$-0.483581\pi$
$19$	0.449490	0.103120	0.0515600	+	0.998670i	$0.483581\pi$
$20$	4.00000	0.894427
$21$	−0.449490	−0.0980867
$22$	−4.89898	−1.04447
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	2.89898	0.568537
$27$	1.00000	0.192450
$28$	−0.449490	−0.0849456
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	4.00000	0.730297
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	−4.89898	−0.852803
$34$	4.89898	0.840168
$35$	−1.79796	−0.303911
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0.449490	0.0729169
$39$	2.89898	0.464208
$40$	4.00000	0.632456
$41$	0.898979	0.140397	0.0701985	−	0.997533i	$-0.477637\pi$
$41$	0.898979	0.140397	0.0701985	+	0.997533i	$0.477637\pi$
$42$	−0.449490	−0.0693578
$43$	7.55051	1.15144	0.575721	−	0.817646i	$-0.304721\pi$
$43$	7.55051	1.15144	0.575721	+	0.817646i	$0.304721\pi$
$44$	−4.89898	−0.738549
$45$	4.00000	0.596285
$46$	−1.00000	−0.147442
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	−6.79796	−0.971137
$50$	11.0000	1.55563
$51$	4.89898	0.685994
$52$	2.89898	0.402016
$53$	0.898979	0.123484	0.0617422	−	0.998092i	$-0.480334\pi$
$53$	0.898979	0.123484	0.0617422	+	0.998092i	$0.480334\pi$
$54$	1.00000	0.136083
$55$	−19.5959	−2.64231
$56$	−0.449490	−0.0600656
$57$	0.449490	0.0595364
$58$	1.00000	0.131306
$59$	4.89898	0.637793	0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898	0.637793	0.318896	+	0.947790i	$0.396688\pi$
$60$	4.00000	0.516398
$61$	−6.89898	−0.883324	−0.441662	−	0.897182i	$-0.645611\pi$
$61$	−6.89898	−0.883324	−0.441662	+	0.897182i	$0.645611\pi$
$62$	−6.00000	−0.762001
$63$	−0.449490	−0.0566304
$64$	1.00000	0.125000
$65$	11.5959	1.43830
$66$	−4.89898	−0.603023
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	4.89898	0.594089
$69$	−1.00000	−0.120386
$70$	−1.79796	−0.214897
$71$	−1.10102	−0.130667	−0.0653335	−	0.997863i	$-0.520811\pi$
$71$	−1.10102	−0.130667	−0.0653335	+	0.997863i	$0.520811\pi$
$72$	1.00000	0.117851
$73$	2.89898	0.339300	0.169650	−	0.985504i	$-0.445736\pi$
$73$	2.89898	0.339300	0.169650	+	0.985504i	$0.445736\pi$
$74$	−2.00000	−0.232495
$75$	11.0000	1.27017
$76$	0.449490	0.0515600
$77$	2.20204	0.250946
$78$	2.89898	0.328245
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	0.898979	0.0992757
$83$	−6.44949	−0.707923	−0.353962	−	0.935260i	$-0.615166\pi$
$83$	−6.44949	−0.707923	−0.353962	+	0.935260i	$0.615166\pi$
$84$	−0.449490	−0.0490434
$85$	19.5959	2.12548
$86$	7.55051	0.814192
$87$	1.00000	0.107211
$88$	−4.89898	−0.522233
$89$	3.10102	0.328708	0.164354	−	0.986401i	$-0.447446\pi$
$89$	3.10102	0.328708	0.164354	+	0.986401i	$0.447446\pi$
$90$	4.00000	0.421637
$91$	−1.30306	−0.136598
$92$	−1.00000	−0.104257
$93$	−6.00000	−0.622171
$94$	4.00000	0.412568
$95$	1.79796	0.184467
$96$	1.00000	0.102062
$97$	15.1464	1.53789	0.768943	−	0.639317i	$-0.220783\pi$
$97$	15.1464	1.53789	0.768943	+	0.639317i	$0.220783\pi$
$98$	−6.79796	−0.686698
$99$	−4.89898	−0.492366
$100$	11.0000	1.10000
$101$	−14.8990	−1.48250	−0.741252	−	0.671227i	$-0.765768\pi$
$101$	−14.8990	−1.48250	−0.741252	+	0.671227i	$0.765768\pi$
$102$	4.89898	0.485071
$103$	−18.2474	−1.79797	−0.898987	−	0.437975i	$-0.855696\pi$
$103$	−18.2474	−1.79797	−0.898987	+	0.437975i	$0.855696\pi$
$104$	2.89898	0.284268
$105$	−1.79796	−0.175463
$106$	0.898979	0.0873166
$107$	−14.4495	−1.39688	−0.698442	−	0.715666i	$-0.746123\pi$
$107$	−14.4495	−1.39688	−0.698442	+	0.715666i	$0.746123\pi$
$108$	1.00000	0.0962250
$109$	0.449490	0.0430533	0.0215267	−	0.999768i	$-0.493147\pi$
$109$	0.449490	0.0430533	0.0215267	+	0.999768i	$0.493147\pi$
$110$	−19.5959	−1.86840
$111$	−2.00000	−0.189832
$112$	−0.449490	−0.0424728
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0.449490	0.0420986
$115$	−4.00000	−0.373002
$116$	1.00000	0.0928477
$117$	2.89898	0.268011
$118$	4.89898	0.450988
$119$	−2.20204	−0.201861
$120$	4.00000	0.365148
$121$	13.0000	1.18182
$122$	−6.89898	−0.624604
$123$	0.898979	0.0810583
$124$	−6.00000	−0.538816
$125$	24.0000	2.14663
$126$	−0.449490	−0.0400437
$127$	−13.7980	−1.22437	−0.612185	−	0.790714i	$-0.709709\pi$
$127$	−13.7980	−1.22437	−0.612185	+	0.790714i	$0.709709\pi$
$128$	1.00000	0.0883883
$129$	7.55051	0.664785
$130$	11.5959	1.01703
$131$	9.10102	0.795160	0.397580	−	0.917568i	$-0.369850\pi$
$131$	9.10102	0.795160	0.397580	+	0.917568i	$0.369850\pi$
$132$	−4.89898	−0.426401
$133$	−0.202041	−0.0175192
$134$	2.00000	0.172774
$135$	4.00000	0.344265
$136$	4.89898	0.420084
$137$	13.7980	1.17884	0.589420	−	0.807827i	$-0.299356\pi$
$137$	13.7980	1.17884	0.589420	+	0.807827i	$0.299356\pi$
$138$	−1.00000	−0.0851257
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	−1.79796	−0.151955
$141$	4.00000	0.336861
$142$	−1.10102	−0.0923956
$143$	−14.2020	−1.18763
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	2.89898	0.239921
$147$	−6.79796	−0.560686
$148$	−2.00000	−0.164399
$149$	10.6969	0.876327	0.438164	−	0.898895i	$-0.355629\pi$
$149$	10.6969	0.876327	0.438164	+	0.898895i	$0.355629\pi$
$150$	11.0000	0.898146
$151$	5.79796	0.471831	0.235916	−	0.971774i	$-0.424191\pi$
$151$	5.79796	0.471831	0.235916	+	0.971774i	$0.424191\pi$
$152$	0.449490	0.0364584
$153$	4.89898	0.396059
$154$	2.20204	0.177446
$155$	−24.0000	−1.92773
$156$	2.89898	0.232104
$157$	3.79796	0.303110	0.151555	−	0.988449i	$-0.451572\pi$
$157$	3.79796	0.303110	0.151555	+	0.988449i	$0.451572\pi$
$158$	10.0000	0.795557
$159$	0.898979	0.0712937
$160$	4.00000	0.316228
$161$	0.449490	0.0354248
$162$	1.00000	0.0785674
$163$	−8.89898	−0.697022	−0.348511	−	0.937305i	$-0.613313\pi$
$163$	−8.89898	−0.697022	−0.348511	+	0.937305i	$0.613313\pi$
$164$	0.898979	0.0701985
$165$	−19.5959	−1.52554
$166$	−6.44949	−0.500577
$167$	−9.79796	−0.758189	−0.379094	−	0.925358i	$-0.623764\pi$
$167$	−9.79796	−0.758189	−0.379094	+	0.925358i	$0.623764\pi$
$168$	−0.449490	−0.0346789
$169$	−4.59592	−0.353532
$170$	19.5959	1.50294
$171$	0.449490	0.0343733
$172$	7.55051	0.575721
$173$	−12.8990	−0.980691	−0.490346	−	0.871528i	$-0.663129\pi$
$173$	−12.8990	−0.980691	−0.490346	+	0.871528i	$0.663129\pi$
$174$	1.00000	0.0758098
$175$	−4.94439	−0.373761
$176$	−4.89898	−0.369274
$177$	4.89898	0.368230
$178$	3.10102	0.232431
$179$	−14.6969	−1.09850	−0.549250	−	0.835658i	$-0.685087\pi$
$179$	−14.6969	−1.09850	−0.549250	+	0.835658i	$0.685087\pi$
$180$	4.00000	0.298142
$181$	−10.6515	−0.791722	−0.395861	−	0.918310i	$-0.629554\pi$
$181$	−10.6515	−0.791722	−0.395861	+	0.918310i	$0.629554\pi$
$182$	−1.30306	−0.0965893
$183$	−6.89898	−0.509987
$184$	−1.00000	−0.0737210
$185$	−8.00000	−0.588172
$186$	−6.00000	−0.439941
$187$	−24.0000	−1.75505
$188$	4.00000	0.291730
$189$	−0.449490	−0.0326956
$190$	1.79796	0.130438
$191$	−7.34847	−0.531717	−0.265858	−	0.964012i	$-0.585655\pi$
$191$	−7.34847	−0.531717	−0.265858	+	0.964012i	$0.585655\pi$
$192$	1.00000	0.0721688
$193$	−17.5959	−1.26658	−0.633291	−	0.773914i	$-0.718296\pi$
$193$	−17.5959	−1.26658	−0.633291	+	0.773914i	$0.718296\pi$
$194$	15.1464	1.08745
$195$	11.5959	0.830401
$196$	−6.79796	−0.485568
$197$	−15.7980	−1.12556	−0.562779	−	0.826607i	$-0.690268\pi$
$197$	−15.7980	−1.12556	−0.562779	+	0.826607i	$0.690268\pi$
$198$	−4.89898	−0.348155
$199$	−1.75255	−0.124235	−0.0621175	−	0.998069i	$-0.519785\pi$
$199$	−1.75255	−0.124235	−0.0621175	+	0.998069i	$0.519785\pi$
$200$	11.0000	0.777817
$201$	2.00000	0.141069
$202$	−14.8990	−1.04829
$203$	−0.449490	−0.0315480
$204$	4.89898	0.342997
$205$	3.59592	0.251150
$206$	−18.2474	−1.27136
$207$	−1.00000	−0.0695048
$208$	2.89898	0.201008
$209$	−2.20204	−0.152318
$210$	−1.79796	−0.124071
$211$	15.1010	1.03960	0.519799	−	0.854289i	$-0.326007\pi$
$211$	15.1010	1.03960	0.519799	+	0.854289i	$0.326007\pi$
$212$	0.898979	0.0617422
$213$	−1.10102	−0.0754407
$214$	−14.4495	−0.987747
$215$	30.2020	2.05976
$216$	1.00000	0.0680414
$217$	2.69694	0.183080
$218$	0.449490	0.0304433
$219$	2.89898	0.195895
$220$	−19.5959	−1.32116
$221$	14.2020	0.955333
$222$	−2.00000	−0.134231
$223$	−5.79796	−0.388260	−0.194130	−	0.980976i	$-0.562188\pi$
$223$	−5.79796	−0.388260	−0.194130	+	0.980976i	$0.562188\pi$
$224$	−0.449490	−0.0300328
$225$	11.0000	0.733333
$226$	0	0
$227$	−7.34847	−0.487735	−0.243868	−	0.969809i	$-0.578416\pi$
$227$	−7.34847	−0.487735	−0.243868	+	0.969809i	$0.578416\pi$
$228$	0.449490	0.0297682
$229$	16.6969	1.10336	0.551682	−	0.834054i	$-0.313986\pi$
$229$	16.6969	1.10336	0.551682	+	0.834054i	$0.313986\pi$
$230$	−4.00000	−0.263752
$231$	2.20204	0.144884
$232$	1.00000	0.0656532
$233$	−12.6969	−0.831804	−0.415902	−	0.909409i	$-0.636534\pi$
$233$	−12.6969	−0.831804	−0.415902	+	0.909409i	$0.636534\pi$
$234$	2.89898	0.189512
$235$	16.0000	1.04372
$236$	4.89898	0.318896
$237$	10.0000	0.649570
$238$	−2.20204	−0.142737
$239$	−10.8990	−0.704996	−0.352498	−	0.935812i	$-0.614668\pi$
$239$	−10.8990	−0.704996	−0.352498	+	0.935812i	$0.614668\pi$
$240$	4.00000	0.258199
$241$	−8.69694	−0.560219	−0.280110	−	0.959968i	$-0.590371\pi$
$241$	−8.69694	−0.560219	−0.280110	+	0.959968i	$0.590371\pi$
$242$	13.0000	0.835672
$243$	1.00000	0.0641500
$244$	−6.89898	−0.441662
$245$	−27.1918	−1.73722
$246$	0.898979	0.0573168
$247$	1.30306	0.0829118
$248$	−6.00000	−0.381000
$249$	−6.44949	−0.408720
$250$	24.0000	1.51789
$251$	12.8990	0.814176	0.407088	−	0.913389i	$-0.366544\pi$
$251$	12.8990	0.814176	0.407088	+	0.913389i	$0.366544\pi$
$252$	−0.449490	−0.0283152
$253$	4.89898	0.307996
$254$	−13.7980	−0.865761
$255$	19.5959	1.22714
$256$	1.00000	0.0625000
$257$	−10.8990	−0.679860	−0.339930	−	0.940451i	$-0.610403\pi$
$257$	−10.8990	−0.679860	−0.339930	+	0.940451i	$0.610403\pi$
$258$	7.55051	0.470074
$259$	0.898979	0.0558599
$260$	11.5959	0.719148
$261$	1.00000	0.0618984
$262$	9.10102	0.562263
$263$	3.75255	0.231392	0.115696	−	0.993285i	$-0.463090\pi$
$263$	3.75255	0.231392	0.115696	+	0.993285i	$0.463090\pi$
$264$	−4.89898	−0.301511
$265$	3.59592	0.220895
$266$	−0.202041	−0.0123879
$267$	3.10102	0.189779
$268$	2.00000	0.122169
$269$	10.8990	0.664523	0.332261	−	0.943187i	$-0.392188\pi$
$269$	10.8990	0.664523	0.332261	+	0.943187i	$0.392188\pi$
$270$	4.00000	0.243432
$271$	−13.7980	−0.838166	−0.419083	−	0.907948i	$-0.637648\pi$
$271$	−13.7980	−0.838166	−0.419083	+	0.907948i	$0.637648\pi$
$272$	4.89898	0.297044
$273$	−1.30306	−0.0788649
$274$	13.7980	0.833565
$275$	−53.8888	−3.24962
$276$	−1.00000	−0.0601929
$277$	24.6969	1.48390	0.741948	−	0.670458i	$-0.233902\pi$
$277$	24.6969	1.48390	0.741948	+	0.670458i	$0.233902\pi$
$278$	6.00000	0.359856
$279$	−6.00000	−0.359211
$280$	−1.79796	−0.107449
$281$	−26.4495	−1.57784	−0.788922	−	0.614493i	$-0.789361\pi$
$281$	−26.4495	−1.57784	−0.788922	+	0.614493i	$0.789361\pi$
$282$	4.00000	0.238197
$283$	−21.5959	−1.28374	−0.641872	−	0.766812i	$-0.721842\pi$
$283$	−21.5959	−1.28374	−0.641872	+	0.766812i	$0.721842\pi$
$284$	−1.10102	−0.0653335
$285$	1.79796	0.106502
$286$	−14.2020	−0.839784
$287$	−0.404082	−0.0238522
$288$	1.00000	0.0589256
$289$	7.00000	0.411765
$290$	4.00000	0.234888
$291$	15.1464	0.887899
$292$	2.89898	0.169650
$293$	−9.14643	−0.534340	−0.267170	−	0.963649i	$-0.586089\pi$
$293$	−9.14643	−0.534340	−0.267170	+	0.963649i	$0.586089\pi$
$294$	−6.79796	−0.396465
$295$	19.5959	1.14092
$296$	−2.00000	−0.116248
$297$	−4.89898	−0.284268
$298$	10.6969	0.619657
$299$	−2.89898	−0.167652
$300$	11.0000	0.635085
$301$	−3.39388	−0.195620
$302$	5.79796	0.333635
$303$	−14.8990	−0.855924
$304$	0.449490	0.0257800
$305$	−27.5959	−1.58014
$306$	4.89898	0.280056
$307$	6.69694	0.382214	0.191107	−	0.981569i	$-0.438792\pi$
$307$	6.69694	0.382214	0.191107	+	0.981569i	$0.438792\pi$
$308$	2.20204	0.125473
$309$	−18.2474	−1.03806
$310$	−24.0000	−1.36311
$311$	26.6969	1.51384	0.756922	−	0.653505i	$-0.226702\pi$
$311$	26.6969	1.51384	0.756922	+	0.653505i	$0.226702\pi$
$312$	2.89898	0.164122
$313$	17.5959	0.994580	0.497290	−	0.867584i	$-0.334328\pi$
$313$	17.5959	0.994580	0.497290	+	0.867584i	$0.334328\pi$
$314$	3.79796	0.214331
$315$	−1.79796	−0.101304
$316$	10.0000	0.562544
$317$	1.10102	0.0618395	0.0309197	−	0.999522i	$-0.490156\pi$
$317$	1.10102	0.0618395	0.0309197	+	0.999522i	$0.490156\pi$
$318$	0.898979	0.0504123
$319$	−4.89898	−0.274290
$320$	4.00000	0.223607
$321$	−14.4495	−0.806492
$322$	0.449490	0.0250491
$323$	2.20204	0.122525
$324$	1.00000	0.0555556
$325$	31.8888	1.76887
$326$	−8.89898	−0.492869
$327$	0.449490	0.0248568
$328$	0.898979	0.0496378
$329$	−1.79796	−0.0991247
$330$	−19.5959	−1.07872
$331$	−20.8990	−1.14871	−0.574356	−	0.818606i	$-0.694747\pi$
$331$	−20.8990	−1.14871	−0.574356	+	0.818606i	$0.694747\pi$
$332$	−6.44949	−0.353962
$333$	−2.00000	−0.109599
$334$	−9.79796	−0.536120
$335$	8.00000	0.437087
$336$	−0.449490	−0.0245217
$337$	15.5505	0.847090	0.423545	−	0.905875i	$-0.360786\pi$
$337$	15.5505	0.847090	0.423545	+	0.905875i	$0.360786\pi$
$338$	−4.59592	−0.249985
$339$	0	0
$340$	19.5959	1.06274
$341$	29.3939	1.59177
$342$	0.449490	0.0243056
$343$	6.20204	0.334879
$344$	7.55051	0.407096
$345$	−4.00000	−0.215353
$346$	−12.8990	−0.693453
$347$	19.5959	1.05196	0.525982	−	0.850496i	$-0.323698\pi$
$347$	19.5959	1.05196	0.525982	+	0.850496i	$0.323698\pi$
$348$	1.00000	0.0536056
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−4.94439	−0.264289
$351$	2.89898	0.154736
$352$	−4.89898	−0.261116
$353$	15.7980	0.840841	0.420420	−	0.907329i	$-0.361883\pi$
$353$	15.7980	0.840841	0.420420	+	0.907329i	$0.361883\pi$
$354$	4.89898	0.260378
$355$	−4.40408	−0.233744
$356$	3.10102	0.164354
$357$	−2.20204	−0.116544
$358$	−14.6969	−0.776757
$359$	−5.55051	−0.292945	−0.146472	−	0.989215i	$-0.546792\pi$
$359$	−5.55051	−0.292945	−0.146472	+	0.989215i	$0.546792\pi$
$360$	4.00000	0.210819
$361$	−18.7980	−0.989366
$362$	−10.6515	−0.559832
$363$	13.0000	0.682323
$364$	−1.30306	−0.0682990
$365$	11.5959	0.606958
$366$	−6.89898	−0.360615
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	−1.00000	−0.0521286
$369$	0.898979	0.0467990
$370$	−8.00000	−0.415900
$371$	−0.404082	−0.0209789
$372$	−6.00000	−0.311086
$373$	−20.0454	−1.03791	−0.518956	−	0.854801i	$-0.673679\pi$
$373$	−20.0454	−1.03791	−0.518956	+	0.854801i	$0.673679\pi$
$374$	−24.0000	−1.24101
$375$	24.0000	1.23935
$376$	4.00000	0.206284
$377$	2.89898	0.149305
$378$	−0.449490	−0.0231193
$379$	17.3485	0.891131	0.445566	−	0.895249i	$-0.353003\pi$
$379$	17.3485	0.891131	0.445566	+	0.895249i	$0.353003\pi$
$380$	1.79796	0.0922333
$381$	−13.7980	−0.706891
$382$	−7.34847	−0.375980
$383$	−9.79796	−0.500652	−0.250326	−	0.968162i	$-0.580538\pi$
$383$	−9.79796	−0.500652	−0.250326	+	0.968162i	$0.580538\pi$
$384$	1.00000	0.0510310
$385$	8.80816	0.448906
$386$	−17.5959	−0.895609
$387$	7.55051	0.383814
$388$	15.1464	0.768943
$389$	31.3485	1.58943	0.794715	−	0.606982i	$-0.207620\pi$
$389$	31.3485	1.58943	0.794715	+	0.606982i	$0.207620\pi$
$390$	11.5959	0.587182
$391$	−4.89898	−0.247752
$392$	−6.79796	−0.343349
$393$	9.10102	0.459086
$394$	−15.7980	−0.795890
$395$	40.0000	2.01262
$396$	−4.89898	−0.246183
$397$	−13.5959	−0.682360	−0.341180	−	0.939998i	$-0.610826\pi$
$397$	−13.5959	−0.682360	−0.341180	+	0.939998i	$0.610826\pi$
$398$	−1.75255	−0.0878475
$399$	−0.202041	−0.0101147
$400$	11.0000	0.550000
$401$	32.2474	1.61036	0.805180	−	0.593030i	$-0.202068\pi$
$401$	32.2474	1.61036	0.805180	+	0.593030i	$0.202068\pi$
$402$	2.00000	0.0997509
$403$	−17.3939	−0.866451
$404$	−14.8990	−0.741252
$405$	4.00000	0.198762
$406$	−0.449490	−0.0223078
$407$	9.79796	0.485667
$408$	4.89898	0.242536
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	3.59592	0.177590
$411$	13.7980	0.680603
$412$	−18.2474	−0.898987
$413$	−2.20204	−0.108355
$414$	−1.00000	−0.0491473
$415$	−25.7980	−1.26637
$416$	2.89898	0.142134
$417$	6.00000	0.293821
$418$	−2.20204	−0.107705
$419$	−32.2474	−1.57539	−0.787695	−	0.616065i	$-0.788726\pi$
$419$	−32.2474	−1.57539	−0.787695	+	0.616065i	$0.788726\pi$
$420$	−1.79796	−0.0877314
$421$	−14.4949	−0.706438	−0.353219	−	0.935541i	$-0.614913\pi$
$421$	−14.4949	−0.706438	−0.353219	+	0.935541i	$0.614913\pi$
$422$	15.1010	0.735106
$423$	4.00000	0.194487
$424$	0.898979	0.0436583
$425$	53.8888	2.61399
$426$	−1.10102	−0.0533446
$427$	3.10102	0.150069
$428$	−14.4495	−0.698442
$429$	−14.2020	−0.685681
$430$	30.2020	1.45647
$431$	36.4949	1.75790	0.878949	−	0.476916i	$-0.158246\pi$
$431$	36.4949	1.75790	0.878949	+	0.476916i	$0.158246\pi$
$432$	1.00000	0.0481125
$433$	26.2474	1.26137	0.630686	−	0.776038i	$-0.282774\pi$
$433$	26.2474	1.26137	0.630686	+	0.776038i	$0.282774\pi$
$434$	2.69694	0.129457
$435$	4.00000	0.191785
$436$	0.449490	0.0215267
$437$	−0.449490	−0.0215020
$438$	2.89898	0.138519
$439$	21.7980	1.04036	0.520180	−	0.854057i	$-0.325865\pi$
$439$	21.7980	1.04036	0.520180	+	0.854057i	$0.325865\pi$
$440$	−19.5959	−0.934199
$441$	−6.79796	−0.323712
$442$	14.2020	0.675522
$443$	18.8990	0.897918	0.448959	−	0.893552i	$-0.351795\pi$
$443$	18.8990	0.897918	0.448959	+	0.893552i	$0.351795\pi$
$444$	−2.00000	−0.0949158
$445$	12.4041	0.588010
$446$	−5.79796	−0.274541
$447$	10.6969	0.505948
$448$	−0.449490	−0.0212364
$449$	−20.4949	−0.967214	−0.483607	−	0.875285i	$-0.660674\pi$
$449$	−20.4949	−0.967214	−0.483607	+	0.875285i	$0.660674\pi$
$450$	11.0000	0.518545
$451$	−4.40408	−0.207380
$452$	0	0
$453$	5.79796	0.272412
$454$	−7.34847	−0.344881
$455$	−5.21225	−0.244354
$456$	0.449490	0.0210493
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	16.6969	0.780197
$459$	4.89898	0.228665
$460$	−4.00000	−0.186501
$461$	8.69694	0.405057	0.202528	−	0.979276i	$-0.435084\pi$
$461$	8.69694	0.405057	0.202528	+	0.979276i	$0.435084\pi$
$462$	2.20204	0.102448
$463$	−29.7980	−1.38483	−0.692414	−	0.721500i	$-0.743453\pi$
$463$	−29.7980	−1.38483	−0.692414	+	0.721500i	$0.743453\pi$
$464$	1.00000	0.0464238
$465$	−24.0000	−1.11297
$466$	−12.6969	−0.588174
$467$	17.7980	0.823591	0.411796	−	0.911276i	$-0.364902\pi$
$467$	17.7980	0.823591	0.411796	+	0.911276i	$0.364902\pi$
$468$	2.89898	0.134005
$469$	−0.898979	−0.0415110
$470$	16.0000	0.738025
$471$	3.79796	0.175001
$472$	4.89898	0.225494
$473$	−36.9898	−1.70079
$474$	10.0000	0.459315
$475$	4.94439	0.226864
$476$	−2.20204	−0.100930
$477$	0.898979	0.0411614
$478$	−10.8990	−0.498508
$479$	24.2474	1.10789	0.553947	−	0.832552i	$-0.313121\pi$
$479$	24.2474	1.10789	0.553947	+	0.832552i	$0.313121\pi$
$480$	4.00000	0.182574
$481$	−5.79796	−0.264364
$482$	−8.69694	−0.396135
$483$	0.449490	0.0204525
$484$	13.0000	0.590909
$485$	60.5857	2.75106
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−6.89898	−0.312302
$489$	−8.89898	−0.402426
$490$	−27.1918	−1.22840
$491$	−5.10102	−0.230206	−0.115103	−	0.993354i	$-0.536720\pi$
$491$	−5.10102	−0.230206	−0.115103	+	0.993354i	$0.536720\pi$
$492$	0.898979	0.0405291
$493$	4.89898	0.220639
$494$	1.30306	0.0586275
$495$	−19.5959	−0.880771
$496$	−6.00000	−0.269408
$497$	0.494897	0.0221992
$498$	−6.44949	−0.289009
$499$	3.79796	0.170020	0.0850100	−	0.996380i	$-0.472908\pi$
$499$	3.79796	0.170020	0.0850100	+	0.996380i	$0.472908\pi$
$500$	24.0000	1.07331
$501$	−9.79796	−0.437741
$502$	12.8990	0.575710
$503$	35.3485	1.57611	0.788055	−	0.615605i	$-0.211088\pi$
$503$	35.3485	1.57611	0.788055	+	0.615605i	$0.211088\pi$
$504$	−0.449490	−0.0200219
$505$	−59.5959	−2.65198
$506$	4.89898	0.217786
$507$	−4.59592	−0.204112
$508$	−13.7980	−0.612185
$509$	12.8990	0.571737	0.285869	−	0.958269i	$-0.407718\pi$
$509$	12.8990	0.571737	0.285869	+	0.958269i	$0.407718\pi$
$510$	19.5959	0.867722
$511$	−1.30306	−0.0576440
$512$	1.00000	0.0441942
$513$	0.449490	0.0198455
$514$	−10.8990	−0.480733
$515$	−72.9898	−3.21631
$516$	7.55051	0.332393
$517$	−19.5959	−0.861827
$518$	0.898979	0.0394989
$519$	−12.8990	−0.566202
$520$	11.5959	0.508515
$521$	18.4495	0.808287	0.404143	−	0.914696i	$-0.367570\pi$
$521$	18.4495	0.808287	0.404143	+	0.914696i	$0.367570\pi$
$522$	1.00000	0.0437688
$523$	19.7980	0.865704	0.432852	−	0.901465i	$-0.357507\pi$
$523$	19.7980	0.865704	0.432852	+	0.901465i	$0.357507\pi$
$524$	9.10102	0.397580
$525$	−4.94439	−0.215791
$526$	3.75255	0.163619
$527$	−29.3939	−1.28042
$528$	−4.89898	−0.213201
$529$	1.00000	0.0434783
$530$	3.59592	0.156197
$531$	4.89898	0.212598
$532$	−0.202041	−0.00875959
$533$	2.60612	0.112884
$534$	3.10102	0.134194
$535$	−57.7980	−2.49882
$536$	2.00000	0.0863868
$537$	−14.6969	−0.634220
$538$	10.8990	0.469888
$539$	33.3031	1.43446
$540$	4.00000	0.172133
$541$	2.20204	0.0946731	0.0473366	−	0.998879i	$-0.484927\pi$
$541$	2.20204	0.0946731	0.0473366	+	0.998879i	$0.484927\pi$
$542$	−13.7980	−0.592673
$543$	−10.6515	−0.457101
$544$	4.89898	0.210042
$545$	1.79796	0.0770161
$546$	−1.30306	−0.0557659
$547$	27.7980	1.18855	0.594277	−	0.804260i	$-0.297438\pi$
$547$	27.7980	1.18855	0.594277	+	0.804260i	$0.297438\pi$
$548$	13.7980	0.589420
$549$	−6.89898	−0.294441
$550$	−53.8888	−2.29783
$551$	0.449490	0.0191489
$552$	−1.00000	−0.0425628
$553$	−4.49490	−0.191142
$554$	24.6969	1.04927
$555$	−8.00000	−0.339581
$556$	6.00000	0.254457
$557$	−21.3939	−0.906488	−0.453244	−	0.891387i	$-0.649733\pi$
$557$	−21.3939	−0.906488	−0.453244	+	0.891387i	$0.649733\pi$
$558$	−6.00000	−0.254000
$559$	21.8888	0.925797
$560$	−1.79796	−0.0759776
$561$	−24.0000	−1.01328
$562$	−26.4495	−1.11570
$563$	7.10102	0.299272	0.149636	−	0.988741i	$-0.452190\pi$
$563$	7.10102	0.299272	0.149636	+	0.988741i	$0.452190\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−21.5959	−0.907744
$567$	−0.449490	−0.0188768
$568$	−1.10102	−0.0461978
$569$	1.79796	0.0753744	0.0376872	−	0.999290i	$-0.488001\pi$
$569$	1.79796	0.0753744	0.0376872	+	0.999290i	$0.488001\pi$
$570$	1.79796	0.0753082
$571$	−34.8990	−1.46048	−0.730238	−	0.683192i	$-0.760591\pi$
$571$	−34.8990	−1.46048	−0.730238	+	0.683192i	$0.760591\pi$
$572$	−14.2020	−0.593817
$573$	−7.34847	−0.306987
$574$	−0.404082	−0.0168661
$575$	−11.0000	−0.458732
$576$	1.00000	0.0416667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	7.00000	0.291162
$579$	−17.5959	−0.731261
$580$	4.00000	0.166091
$581$	2.89898	0.120270
$582$	15.1464	0.627840
$583$	−4.40408	−0.182398
$584$	2.89898	0.119961
$585$	11.5959	0.479432
$586$	−9.14643	−0.377835
$587$	47.1918	1.94782	0.973908	−	0.226944i	$-0.0728735\pi$
$587$	47.1918	1.94782	0.973908	+	0.226944i	$0.0728735\pi$
$588$	−6.79796	−0.280343
$589$	−2.69694	−0.111125
$590$	19.5959	0.806751
$591$	−15.7980	−0.649841
$592$	−2.00000	−0.0821995
$593$	−17.5959	−0.722578	−0.361289	−	0.932454i	$-0.617663\pi$
$593$	−17.5959	−0.722578	−0.361289	+	0.932454i	$0.617663\pi$
$594$	−4.89898	−0.201008
$595$	−8.80816	−0.361100
$596$	10.6969	0.438164
$597$	−1.75255	−0.0717271
$598$	−2.89898	−0.118548
$599$	−6.69694	−0.273629	−0.136815	−	0.990597i	$-0.543687\pi$
$599$	−6.69694	−0.273629	−0.136815	+	0.990597i	$0.543687\pi$
$600$	11.0000	0.449073
$601$	−19.3939	−0.791093	−0.395546	−	0.918446i	$-0.629445\pi$
$601$	−19.3939	−0.791093	−0.395546	+	0.918446i	$0.629445\pi$
$602$	−3.39388	−0.138324
$603$	2.00000	0.0814463
$604$	5.79796	0.235916
$605$	52.0000	2.11410
$606$	−14.8990	−0.605230
$607$	−3.79796	−0.154154	−0.0770772	−	0.997025i	$-0.524559\pi$
$607$	−3.79796	−0.154154	−0.0770772	+	0.997025i	$0.524559\pi$
$608$	0.449490	0.0182292
$609$	−0.449490	−0.0182142
$610$	−27.5959	−1.11733
$611$	11.5959	0.469121
$612$	4.89898	0.198030
$613$	36.0454	1.45586	0.727930	−	0.685651i	$-0.240482\pi$
$613$	36.0454	1.45586	0.727930	+	0.685651i	$0.240482\pi$
$614$	6.69694	0.270266
$615$	3.59592	0.145001
$616$	2.20204	0.0887228
$617$	−19.5959	−0.788902	−0.394451	−	0.918917i	$-0.629065\pi$
$617$	−19.5959	−0.788902	−0.394451	+	0.918917i	$0.629065\pi$
$618$	−18.2474	−0.734020
$619$	−24.0454	−0.966467	−0.483233	−	0.875492i	$-0.660538\pi$
$619$	−24.0454	−0.966467	−0.483233	+	0.875492i	$0.660538\pi$
$620$	−24.0000	−0.963863
$621$	−1.00000	−0.0401286
$622$	26.6969	1.07045
$623$	−1.39388	−0.0558445
$624$	2.89898	0.116052
$625$	41.0000	1.64000
$626$	17.5959	0.703274
$627$	−2.20204	−0.0879410
$628$	3.79796	0.151555
$629$	−9.79796	−0.390670
$630$	−1.79796	−0.0716324
$631$	−8.44949	−0.336369	−0.168184	−	0.985756i	$-0.553790\pi$
$631$	−8.44949	−0.336369	−0.168184	+	0.985756i	$0.553790\pi$
$632$	10.0000	0.397779
$633$	15.1010	0.600212
$634$	1.10102	0.0437271
$635$	−55.1918	−2.19022
$636$	0.898979	0.0356469
$637$	−19.7071	−0.780825
$638$	−4.89898	−0.193952
$639$	−1.10102	−0.0435557
$640$	4.00000	0.158114
$641$	−0.404082	−0.0159603	−0.00798014	−	0.999968i	$-0.502540\pi$
$641$	−0.404082	−0.0159603	−0.00798014	+	0.999968i	$0.502540\pi$
$642$	−14.4495	−0.570276
$643$	−47.7980	−1.88497	−0.942484	−	0.334252i	$-0.891516\pi$
$643$	−47.7980	−1.88497	−0.942484	+	0.334252i	$0.891516\pi$
$644$	0.449490	0.0177124
$645$	30.2020	1.18920
$646$	2.20204	0.0866381
$647$	−14.8990	−0.585739	−0.292870	−	0.956152i	$-0.594610\pi$
$647$	−14.8990	−0.585739	−0.292870	+	0.956152i	$0.594610\pi$
$648$	1.00000	0.0392837
$649$	−24.0000	−0.942082
$650$	31.8888	1.25078
$651$	2.69694	0.105701
$652$	−8.89898	−0.348511
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0.449490	0.0175764
$655$	36.4041	1.42243
$656$	0.898979	0.0350993
$657$	2.89898	0.113100
$658$	−1.79796	−0.0700917
$659$	−43.5959	−1.69826	−0.849128	−	0.528187i	$-0.822872\pi$
$659$	−43.5959	−1.69826	−0.849128	+	0.528187i	$0.822872\pi$
$660$	−19.5959	−0.762770
$661$	−33.3485	−1.29711	−0.648553	−	0.761170i	$-0.724625\pi$
$661$	−33.3485	−1.29711	−0.648553	+	0.761170i	$0.724625\pi$
$662$	−20.8990	−0.812262
$663$	14.2020	0.551562
$664$	−6.44949	−0.250289
$665$	−0.808164	−0.0313393
$666$	−2.00000	−0.0774984
$667$	−1.00000	−0.0387202
$668$	−9.79796	−0.379094
$669$	−5.79796	−0.224162
$670$	8.00000	0.309067
$671$	33.7980	1.30476
$672$	−0.449490	−0.0173394
$673$	−41.5959	−1.60340	−0.801702	−	0.597723i	$-0.796072\pi$
$673$	−41.5959	−1.60340	−0.801702	+	0.597723i	$0.796072\pi$
$674$	15.5505	0.598983
$675$	11.0000	0.423390
$676$	−4.59592	−0.176766
$677$	−4.65153	−0.178773	−0.0893864	−	0.995997i	$-0.528491\pi$
$677$	−4.65153	−0.178773	−0.0893864	+	0.995997i	$0.528491\pi$
$678$	0	0
$679$	−6.80816	−0.261273
$680$	19.5959	0.751469
$681$	−7.34847	−0.281594
$682$	29.3939	1.12555
$683$	−19.5959	−0.749817	−0.374908	−	0.927062i	$-0.622326\pi$
$683$	−19.5959	−0.749817	−0.374908	+	0.927062i	$0.622326\pi$
$684$	0.449490	0.0171867
$685$	55.1918	2.10877
$686$	6.20204	0.236795
$687$	16.6969	0.637028
$688$	7.55051	0.287861
$689$	2.60612	0.0992854
$690$	−4.00000	−0.152277
$691$	29.3939	1.11820	0.559098	−	0.829102i	$-0.311148\pi$
$691$	29.3939	1.11820	0.559098	+	0.829102i	$0.311148\pi$
$692$	−12.8990	−0.490346
$693$	2.20204	0.0836486
$694$	19.5959	0.743851
$695$	24.0000	0.910372
$696$	1.00000	0.0379049
$697$	4.40408	0.166817
$698$	14.0000	0.529908
$699$	−12.6969	−0.480242
$700$	−4.94439	−0.186880
$701$	6.20204	0.234248	0.117124	−	0.993117i	$-0.462633\pi$
$701$	6.20204	0.234248	0.117124	+	0.993117i	$0.462633\pi$
$702$	2.89898	0.109415
$703$	−0.898979	−0.0339057
$704$	−4.89898	−0.184637
$705$	16.0000	0.602595
$706$	15.7980	0.594564
$707$	6.69694	0.251864
$708$	4.89898	0.184115
$709$	−18.2474	−0.685297	−0.342649	−	0.939464i	$-0.611324\pi$
$709$	−18.2474	−0.685297	−0.342649	+	0.939464i	$0.611324\pi$
$710$	−4.40408	−0.165282
$711$	10.0000	0.375029
$712$	3.10102	0.116216
$713$	6.00000	0.224702
$714$	−2.20204	−0.0824093
$715$	−56.8082	−2.12450
$716$	−14.6969	−0.549250
$717$	−10.8990	−0.407030
$718$	−5.55051	−0.207143
$719$	37.3939	1.39456	0.697278	−	0.716801i	$-0.254394\pi$
$719$	37.3939	1.39456	0.697278	+	0.716801i	$0.254394\pi$
$720$	4.00000	0.149071
$721$	8.20204	0.305460
$722$	−18.7980	−0.699588
$723$	−8.69694	−0.323443
$724$	−10.6515	−0.395861
$725$	11.0000	0.408530
$726$	13.0000	0.482475
$727$	0.202041	0.00749329	0.00374664	−	0.999993i	$-0.498807\pi$
$727$	0.202041	0.00749329	0.00374664	+	0.999993i	$0.498807\pi$
$728$	−1.30306	−0.0482947
$729$	1.00000	0.0370370
$730$	11.5959	0.429184
$731$	36.9898	1.36812
$732$	−6.89898	−0.254994
$733$	−0.202041	−0.00746256	−0.00373128	−	0.999993i	$-0.501188\pi$
$733$	−0.202041	−0.00746256	−0.00373128	+	0.999993i	$0.501188\pi$
$734$	−30.0000	−1.10732
$735$	−27.1918	−1.00299
$736$	−1.00000	−0.0368605
$737$	−9.79796	−0.360912
$738$	0.898979	0.0330919
$739$	−14.2020	−0.522431	−0.261215	−	0.965281i	$-0.584123\pi$
$739$	−14.2020	−0.522431	−0.261215	+	0.965281i	$0.584123\pi$
$740$	−8.00000	−0.294086
$741$	1.30306	0.0478692
$742$	−0.404082	−0.0148343
$743$	−1.14643	−0.0420584	−0.0210292	−	0.999779i	$-0.506694\pi$
$743$	−1.14643	−0.0420584	−0.0210292	+	0.999779i	$0.506694\pi$
$744$	−6.00000	−0.219971
$745$	42.7878	1.56762
$746$	−20.0454	−0.733915
$747$	−6.44949	−0.235974
$748$	−24.0000	−0.877527
$749$	6.49490	0.237318
$750$	24.0000	0.876356
$751$	−11.7980	−0.430514	−0.215257	−	0.976557i	$-0.569059\pi$
$751$	−11.7980	−0.430514	−0.215257	+	0.976557i	$0.569059\pi$
$752$	4.00000	0.145865
$753$	12.8990	0.470065
$754$	2.89898	0.105575
$755$	23.1918	0.844037
$756$	−0.449490	−0.0163478
$757$	−42.4949	−1.54450	−0.772252	−	0.635317i	$-0.780870\pi$
$757$	−42.4949	−1.54450	−0.772252	+	0.635317i	$0.780870\pi$
$758$	17.3485	0.630125
$759$	4.89898	0.177822
$760$	1.79796	0.0652188
$761$	−47.3939	−1.71803	−0.859013	−	0.511953i	$-0.828922\pi$
$761$	−47.3939	−1.71803	−0.859013	+	0.511953i	$0.828922\pi$
$762$	−13.7980	−0.499847
$763$	−0.202041	−0.00731438
$764$	−7.34847	−0.265858
$765$	19.5959	0.708492
$766$	−9.79796	−0.354015
$767$	14.2020	0.512806
$768$	1.00000	0.0360844
$769$	34.2474	1.23499	0.617497	−	0.786573i	$-0.288147\pi$
$769$	34.2474	1.23499	0.617497	+	0.786573i	$0.288147\pi$
$770$	8.80816	0.317424
$771$	−10.8990	−0.392517
$772$	−17.5959	−0.633291
$773$	54.4495	1.95841	0.979206	−	0.202868i	$-0.0650264\pi$
$773$	54.4495	1.95841	0.979206	+	0.202868i	$0.0650264\pi$
$774$	7.55051	0.271397
$775$	−66.0000	−2.37079
$776$	15.1464	0.543725
$777$	0.898979	0.0322507
$778$	31.3485	1.12390
$779$	0.404082	0.0144777
$780$	11.5959	0.415200
$781$	5.39388	0.193008
$782$	−4.89898	−0.175187
$783$	1.00000	0.0357371
$784$	−6.79796	−0.242784
$785$	15.1918	0.542220
$786$	9.10102	0.324623
$787$	34.0000	1.21197	0.605985	−	0.795476i	$-0.292779\pi$
$787$	34.0000	1.21197	0.605985	+	0.795476i	$0.292779\pi$
$788$	−15.7980	−0.562779
$789$	3.75255	0.133594
$790$	40.0000	1.42314
$791$	0	0
$792$	−4.89898	−0.174078
$793$	−20.0000	−0.710221
$794$	−13.5959	−0.482501
$795$	3.59592	0.127534
$796$	−1.75255	−0.0621175
$797$	−39.8434	−1.41132	−0.705662	−	0.708548i	$-0.749350\pi$
$797$	−39.8434	−1.41132	−0.705662	+	0.708548i	$0.749350\pi$
$798$	−0.202041	−0.00715217
$799$	19.5959	0.693254
$800$	11.0000	0.388909
$801$	3.10102	0.109569
$802$	32.2474	1.13870
$803$	−14.2020	−0.501179
$804$	2.00000	0.0705346
$805$	1.79796	0.0633697
$806$	−17.3939	−0.612673
$807$	10.8990	0.383662
$808$	−14.8990	−0.524144
$809$	20.4949	0.720562	0.360281	−	0.932844i	$-0.382681\pi$
$809$	20.4949	0.720562	0.360281	+	0.932844i	$0.382681\pi$
$810$	4.00000	0.140546
$811$	−45.5959	−1.60109	−0.800545	−	0.599273i	$-0.795456\pi$
$811$	−45.5959	−1.60109	−0.800545	+	0.599273i	$0.795456\pi$
$812$	−0.449490	−0.0157740
$813$	−13.7980	−0.483916
$814$	9.79796	0.343418
$815$	−35.5959	−1.24687
$816$	4.89898	0.171499
$817$	3.39388	0.118737
$818$	−6.00000	−0.209785
$819$	−1.30306	−0.0455327
$820$	3.59592	0.125575
$821$	32.8990	1.14818	0.574091	−	0.818791i	$-0.305356\pi$
$821$	32.8990	1.14818	0.574091	+	0.818791i	$0.305356\pi$
$822$	13.7980	0.481259
$823$	−26.2020	−0.913346	−0.456673	−	0.889635i	$-0.650959\pi$
$823$	−26.2020	−0.913346	−0.456673	+	0.889635i	$0.650959\pi$
$824$	−18.2474	−0.635680
$825$	−53.8888	−1.87617
$826$	−2.20204	−0.0766188
$827$	7.59592	0.264136	0.132068	−	0.991241i	$-0.457838\pi$
$827$	7.59592	0.264136	0.132068	+	0.991241i	$0.457838\pi$
$828$	−1.00000	−0.0347524
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−25.7980	−0.895460
$831$	24.6969	0.856727
$832$	2.89898	0.100504
$833$	−33.3031	−1.15388
$834$	6.00000	0.207763
$835$	−39.1918	−1.35629
$836$	−2.20204	−0.0761592
$837$	−6.00000	−0.207390
$838$	−32.2474	−1.11397
$839$	52.7423	1.82087	0.910434	−	0.413654i	$-0.135748\pi$
$839$	52.7423	1.82087	0.910434	+	0.413654i	$0.135748\pi$
$840$	−1.79796	−0.0620355
$841$	1.00000	0.0344828
$842$	−14.4949	−0.499527
$843$	−26.4495	−0.910969
$844$	15.1010	0.519799
$845$	−18.3837	−0.632418
$846$	4.00000	0.137523
$847$	−5.84337	−0.200780
$848$	0.898979	0.0308711
$849$	−21.5959	−0.741170
$850$	53.8888	1.84837
$851$	2.00000	0.0685591
$852$	−1.10102	−0.0377203
$853$	−25.1918	−0.862552	−0.431276	−	0.902220i	$-0.641936\pi$
$853$	−25.1918	−0.862552	−0.431276	+	0.902220i	$0.641936\pi$
$854$	3.10102	0.106115
$855$	1.79796	0.0614889
$856$	−14.4495	−0.493873
$857$	34.8990	1.19213	0.596063	−	0.802938i	$-0.296731\pi$
$857$	34.8990	1.19213	0.596063	+	0.802938i	$0.296731\pi$
$858$	−14.2020	−0.484850
$859$	3.59592	0.122691	0.0613456	−	0.998117i	$-0.480461\pi$
$859$	3.59592	0.122691	0.0613456	+	0.998117i	$0.480461\pi$
$860$	30.2020	1.02988
$861$	−0.404082	−0.0137711
$862$	36.4949	1.24302
$863$	17.7980	0.605850	0.302925	−	0.953014i	$-0.402037\pi$
$863$	17.7980	0.605850	0.302925	+	0.953014i	$0.402037\pi$
$864$	1.00000	0.0340207
$865$	−51.5959	−1.75431
$866$	26.2474	0.891925
$867$	7.00000	0.237732
$868$	2.69694	0.0915401
$869$	−48.9898	−1.66186
$870$	4.00000	0.135613
$871$	5.79796	0.196456
$872$	0.449490	0.0152216
$873$	15.1464	0.512629
$874$	−0.449490	−0.0152042
$875$	−10.7878	−0.364693
$876$	2.89898	0.0979474
$877$	12.6969	0.428745	0.214373	−	0.976752i	$-0.431229\pi$
$877$	12.6969	0.428745	0.214373	+	0.976752i	$0.431229\pi$
$878$	21.7980	0.735645
$879$	−9.14643	−0.308501
$880$	−19.5959	−0.660578
$881$	−11.1010	−0.374003	−0.187001	−	0.982360i	$-0.559877\pi$
$881$	−11.1010	−0.374003	−0.187001	+	0.982360i	$0.559877\pi$
$882$	−6.79796	−0.228899
$883$	−17.7980	−0.598949	−0.299475	−	0.954104i	$-0.596811\pi$
$883$	−17.7980	−0.598949	−0.299475	+	0.954104i	$0.596811\pi$
$884$	14.2020	0.477666
$885$	19.5959	0.658710
$886$	18.8990	0.634924
$887$	−4.89898	−0.164492	−0.0822458	−	0.996612i	$-0.526209\pi$
$887$	−4.89898	−0.164492	−0.0822458	+	0.996612i	$0.526209\pi$
$888$	−2.00000	−0.0671156
$889$	6.20204	0.208010
$890$	12.4041	0.415786
$891$	−4.89898	−0.164122
$892$	−5.79796	−0.194130
$893$	1.79796	0.0601664
$894$	10.6969	0.357759
$895$	−58.7878	−1.96506
$896$	−0.449490	−0.0150164
$897$	−2.89898	−0.0967941
$898$	−20.4949	−0.683924
$899$	−6.00000	−0.200111
$900$	11.0000	0.366667
$901$	4.40408	0.146721
$902$	−4.40408	−0.146640
$903$	−3.39388	−0.112941
$904$	0	0
$905$	−42.6061	−1.41628
$906$	5.79796	0.192624
$907$	25.3485	0.841682	0.420841	−	0.907134i	$-0.361735\pi$
$907$	25.3485	0.841682	0.420841	+	0.907134i	$0.361735\pi$
$908$	−7.34847	−0.243868
$909$	−14.8990	−0.494168
$910$	−5.21225	−0.172784
$911$	−27.8434	−0.922492	−0.461246	−	0.887272i	$-0.652597\pi$
$911$	−27.8434	−0.922492	−0.461246	+	0.887272i	$0.652597\pi$
$912$	0.449490	0.0148841
$913$	31.5959	1.04567
$914$	−22.0000	−0.727695
$915$	−27.5959	−0.912293
$916$	16.6969	0.551682
$917$	−4.09082	−0.135091
$918$	4.89898	0.161690
$919$	−26.2474	−0.865823	−0.432912	−	0.901436i	$-0.642514\pi$
$919$	−26.2474	−0.865823	−0.432912	+	0.901436i	$0.642514\pi$
$920$	−4.00000	−0.131876
$921$	6.69694	0.220672
$922$	8.69694	0.286418
$923$	−3.19184	−0.105061
$924$	2.20204	0.0724418
$925$	−22.0000	−0.723356
$926$	−29.7980	−0.979222
$927$	−18.2474	−0.599325
$928$	1.00000	0.0328266
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	−24.0000	−0.786991
$931$	−3.05561	−0.100144
$932$	−12.6969	−0.415902
$933$	26.6969	0.874019
$934$	17.7980	0.582367
$935$	−96.0000	−3.13954
$936$	2.89898	0.0947561
$937$	7.79796	0.254748	0.127374	−	0.991855i	$-0.459345\pi$
$937$	7.79796	0.254748	0.127374	+	0.991855i	$0.459345\pi$
$938$	−0.898979	−0.0293527
$939$	17.5959	0.574221
$940$	16.0000	0.521862
$941$	35.1918	1.14722	0.573611	−	0.819128i	$-0.305542\pi$
$941$	35.1918	1.14722	0.573611	+	0.819128i	$0.305542\pi$
$942$	3.79796	0.123744
$943$	−0.898979	−0.0292748
$944$	4.89898	0.159448
$945$	−1.79796	−0.0584876
$946$	−36.9898	−1.20264
$947$	9.10102	0.295743	0.147872	−	0.989007i	$-0.452758\pi$
$947$	9.10102	0.295743	0.147872	+	0.989007i	$0.452758\pi$
$948$	10.0000	0.324785
$949$	8.40408	0.272808
$950$	4.94439	0.160417
$951$	1.10102	0.0357030
$952$	−2.20204	−0.0713686
$953$	−54.9444	−1.77982	−0.889912	−	0.456133i	$-0.849234\pi$
$953$	−54.9444	−1.77982	−0.889912	+	0.456133i	$0.849234\pi$
$954$	0.898979	0.0291055
$955$	−29.3939	−0.951164
$956$	−10.8990	−0.352498
$957$	−4.89898	−0.158362
$958$	24.2474	0.783400
$959$	−6.20204	−0.200274
$960$	4.00000	0.129099
$961$	5.00000	0.161290
$962$	−5.79796	−0.186934
$963$	−14.4495	−0.465628
$964$	−8.69694	−0.280110
$965$	−70.3837	−2.26573
$966$	0.449490	0.0144621
$967$	57.3939	1.84566	0.922831	−	0.385204i	$-0.125869\pi$
$967$	57.3939	1.84566	0.922831	+	0.385204i	$0.125869\pi$
$968$	13.0000	0.417836
$969$	2.20204	0.0707397
$970$	60.5857	1.94529
$971$	38.2929	1.22888	0.614438	−	0.788965i	$-0.289383\pi$
$971$	38.2929	1.22888	0.614438	+	0.788965i	$0.289383\pi$
$972$	1.00000	0.0320750
$973$	−2.69694	−0.0864599
$974$	−32.0000	−1.02535
$975$	31.8888	1.02126
$976$	−6.89898	−0.220831
$977$	−44.7423	−1.43143	−0.715717	−	0.698390i	$-0.753900\pi$
$977$	−44.7423	−1.43143	−0.715717	+	0.698390i	$0.753900\pi$
$978$	−8.89898	−0.284558
$979$	−15.1918	−0.485533
$980$	−27.1918	−0.868611
$981$	0.449490	0.0143511
$982$	−5.10102	−0.162780
$983$	−14.9444	−0.476652	−0.238326	−	0.971185i	$-0.576599\pi$
$983$	−14.9444	−0.476652	−0.238326	+	0.971185i	$0.576599\pi$
$984$	0.898979	0.0286584
$985$	−63.1918	−2.01346
$986$	4.89898	0.156015
$987$	−1.79796	−0.0572297
$988$	1.30306	0.0414559
$989$	−7.55051	−0.240092
$990$	−19.5959	−0.622799
$991$	−43.1010	−1.36915	−0.684575	−	0.728943i	$-0.740012\pi$
$991$	−43.1010	−1.36915	−0.684575	+	0.728943i	$0.740012\pi$
$992$	−6.00000	−0.190500
$993$	−20.8990	−0.663209
$994$	0.494897	0.0156972
$995$	−7.01021	−0.222238
$996$	−6.44949	−0.204360
$997$	−1.59592	−0.0505432	−0.0252716	−	0.999681i	$-0.508045\pi$
$997$	−1.59592	−0.0505432	−0.0252716	+	0.999681i	$0.508045\pi$
$998$	3.79796	0.120222
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.x.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.x.1.1	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -0.449490 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -0.449490 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -4.89898 q^{11} +1.00000 q^{12} +2.89898 q^{13} -0.449490 q^{14} +4.00000 q^{15} +1.00000 q^{16} +4.89898 q^{17} +1.00000 q^{18} +0.449490 q^{19} +4.00000 q^{20} -0.449490 q^{21} -4.89898 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +2.89898 q^{26} +1.00000 q^{27} -0.449490 q^{28} +1.00000 q^{29} +4.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} -4.89898 q^{33} +4.89898 q^{34} -1.79796 q^{35} +1.00000 q^{36} -2.00000 q^{37} +0.449490 q^{38} +2.89898 q^{39} +4.00000 q^{40} +0.898979 q^{41} -0.449490 q^{42} +7.55051 q^{43} -4.89898 q^{44} +4.00000 q^{45} -1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -6.79796 q^{49} +11.0000 q^{50} +4.89898 q^{51} +2.89898 q^{52} +0.898979 q^{53} +1.00000 q^{54} -19.5959 q^{55} -0.449490 q^{56} +0.449490 q^{57} +1.00000 q^{58} +4.89898 q^{59} +4.00000 q^{60} -6.89898 q^{61} -6.00000 q^{62} -0.449490 q^{63} +1.00000 q^{64} +11.5959 q^{65} -4.89898 q^{66} +2.00000 q^{67} +4.89898 q^{68} -1.00000 q^{69} -1.79796 q^{70} -1.10102 q^{71} +1.00000 q^{72} +2.89898 q^{73} -2.00000 q^{74} +11.0000 q^{75} +0.449490 q^{76} +2.20204 q^{77} +2.89898 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +0.898979 q^{82} -6.44949 q^{83} -0.449490 q^{84} +19.5959 q^{85} +7.55051 q^{86} +1.00000 q^{87} -4.89898 q^{88} +3.10102 q^{89} +4.00000 q^{90} -1.30306 q^{91} -1.00000 q^{92} -6.00000 q^{93} +4.00000 q^{94} +1.79796 q^{95} +1.00000 q^{96} +15.1464 q^{97} -6.79796 q^{98} -4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 8 q^{5} + 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 8 * q^5 + 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 8 q^{5} + 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9} + 8 q^{10} + 2 q^{12} - 4 q^{13} + 4 q^{14} + 8 q^{15} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 8 q^{20} + 4 q^{21} - 2 q^{23} + 2 q^{24} + 22 q^{25} - 4 q^{26} + 2 q^{27} + 4 q^{28} + 2 q^{29} + 8 q^{30} - 12 q^{31} + 2 q^{32} + 16 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} + 8 q^{40} - 8 q^{41} + 4 q^{42} + 20 q^{43} + 8 q^{45} - 2 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{49} + 22 q^{50} - 4 q^{52} - 8 q^{53} + 2 q^{54} + 4 q^{56} - 4 q^{57} + 2 q^{58} + 8 q^{60} - 4 q^{61} - 12 q^{62} + 4 q^{63} + 2 q^{64} - 16 q^{65} + 4 q^{67} - 2 q^{69} + 16 q^{70} - 12 q^{71} + 2 q^{72} - 4 q^{73} - 4 q^{74} + 22 q^{75} - 4 q^{76} + 24 q^{77} - 4 q^{78} + 20 q^{79} + 8 q^{80} + 2 q^{81} - 8 q^{82} - 8 q^{83} + 4 q^{84} + 20 q^{86} + 2 q^{87} + 16 q^{89} + 8 q^{90} - 32 q^{91} - 2 q^{92} - 12 q^{93} + 8 q^{94} - 16 q^{95} + 2 q^{96} - 4 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 8 * q^5 + 2 * q^6 + 4 * q^7 + 2 * q^8 + 2 * q^9 + 8 * q^10 + 2 * q^12 - 4 * q^13 + 4 * q^14 + 8 * q^15 + 2 * q^16 + 2 * q^18 - 4 * q^19 + 8 * q^20 + 4 * q^21 - 2 * q^23 + 2 * q^24 + 22 * q^25 - 4 * q^26 + 2 * q^27 + 4 * q^28 + 2 * q^29 + 8 * q^30 - 12 * q^31 + 2 * q^32 + 16 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 + 8 * q^40 - 8 * q^41 + 4 * q^42 + 20 * q^43 + 8 * q^45 - 2 * q^46 + 8 * q^47 + 2 * q^48 + 6 * q^49 + 22 * q^50 - 4 * q^52 - 8 * q^53 + 2 * q^54 + 4 * q^56 - 4 * q^57 + 2 * q^58 + 8 * q^60 - 4 * q^61 - 12 * q^62 + 4 * q^63 + 2 * q^64 - 16 * q^65 + 4 * q^67 - 2 * q^69 + 16 * q^70 - 12 * q^71 + 2 * q^72 - 4 * q^73 - 4 * q^74 + 22 * q^75 - 4 * q^76 + 24 * q^77 - 4 * q^78 + 20 * q^79 + 8 * q^80 + 2 * q^81 - 8 * q^82 - 8 * q^83 + 4 * q^84 + 20 * q^86 + 2 * q^87 + 16 * q^89 + 8 * q^90 - 32 * q^91 - 2 * q^92 - 12 * q^93 + 8 * q^94 - 16 * q^95 + 2 * q^96 - 4 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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